So you've learned how to solve systems of equations with
graphs, but that takes an awful lot of work, paper, and time. Wouldn't it
be simpler if you could do it without all the hassle (gosh, I feel like I'm
selling a product on a low-budget commercial *_*)?

Well there is a way...two ways actually. The first one, my
personal favorite, is on this page. The other is on the next page.

First you work in one equation, solving for one variable in
terms of another. Let's solve for y in terms of x in the first equation
(you can do whichever variable in whichever equation you want, but you should
experiment, and start recognizing which is the simplest one).

x + y = 4-x -xy = 4 - x

Now this is where the substitution comes in. We substitute
this value for y in the second equation.

Uh-oh, we got a constant equals a constant. It's a true
statement, but where did we go wrong? The answer is: nowhere. If you
remember graphing this (or if you do it on your own), you know that these
equations coincide. There are infinitely many solutions that will satisfy
the system of equations. A rule of thumb (love that phrase!) is that if
you get identity, that is an equation that is always true (constant = constant),
there are infinitely many solutions. The graphs will coincide. These
are called dependent equations. You can sometimes see these at a
glance (for example, each term in one equation is multiplied by two, and that is
the second equations, as in this case).

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How about one more... x + y = 4 and x + y
= 5

x + y = 4
x = 4 - y
x + y = 5
4 - y + y = 5
4 = 5

What??? Now this has to be wrong, you think.
And you are right, it is wrong. But we haven't done anything wrong.
If you get a false statement (a constant = a different constant), the equations
have no common solutions. The lines will be parallel. These are
called inconsistent equations.